We are asked to solve the quadratic equation $x^2 + 10x - 1 = 0$ for $x$. We need to provide an exact answer using radicals if necessary.

1. Problem Description

We are asked to solve the quadratic equation $x^2 + 10x - 1 = 0$ for $x$. We need to provide an exact answer using radicals if necessary.

2. Solution Steps

We can solve the quadratic equation using the quadratic formula. The quadratic formula is given by:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

where $a$, $b$, and $c$ are the coefficients of the quadratic equation $ax^2 + bx + c = 0$.

In our case, $a=1$, $b=10$, and $c=-1$.

Plugging these values into the quadratic formula, we get:

$x = \frac{-10 \pm \sqrt{10^2 - 4(1)(-1)}}{2(1)}$

$x = \frac{-10 \pm \sqrt{100 + 4}}{2}$

$x = \frac{-10 \pm \sqrt{104}}{2}$

We can simplify $\sqrt{104}$ as follows:

$\sqrt{104} = \sqrt{4 \times 26} = \sqrt{4} \times \sqrt{26} = 2\sqrt{26}$

Therefore,

$x = \frac{-10 \pm 2\sqrt{26}}{2}$

We can divide both terms in the numerator by 2:

$x = -5 \pm \sqrt{26}$

Thus, the two solutions are $x = -5 + \sqrt{26}$ and $x = -5 - \sqrt{26}$.

3. Final Answer

$x = -5 + \sqrt{26}, -5 - \sqrt{26}$